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A210565 Triangle of coefficients of polynomials u(n,x) jointly generated with A210595; see the Formula section. 3
1, 2, 2, 3, 5, 3, 4, 9, 10, 5, 5, 14, 22, 20, 8, 6, 20, 40, 51, 38, 13, 7, 27, 65, 105, 111, 71, 21, 8, 35, 98, 190, 256, 233, 130, 34, 9, 44, 140, 315, 511, 594, 474, 235, 55, 10, 54, 192, 490, 924, 1295, 1324, 942, 420, 89, 11, 65, 255, 726, 1554, 2534, 3130, 2860, 1836, 744, 144 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Row n starts with n and ends with F(n+1), where F=A000045 (Fibonacci numbers).

Row sums: A005409.

Alternating row sums: 1,0,1,0,1,0,1,0,1,0,1,0, ...

For a discussion and guide to related arrays, see A208510.

LINKS

G. C. Greubel, Rows n = 1..30 of the triangle, flattened

FORMULA

u(n,x) = x*u(n-1,x) + (x+1)*v(n-1,x) + 1,

v(n,x) = x*u(n-1,x) + v(n-1,x) + 1,

where u(1,x) = 1, v(1,x) = 1.

T(n, k) = [x^k]( u(n, x) ), where u(n, x) = (1+x)*u(n-1,x) + x^2*u(n-2,x) + 1 + x, u(1, x) = 1, and u(2, x) = 2 + 2*x. - G. C. Greubel, May 24 2021

EXAMPLE

First five rows:

  1;

  2,  2;

  3,  5,  3;

  4,  9, 10,  5;

  5, 14, 22, 20, 8;

First three polynomials u(n,x):

u(1, x) = 1;

u(2, x) = 2 + 2*x;

u(3, x) = 3 + 5*x + 3*x^2.

MATHEMATICA

(* First program *)

u[1, x_]:= 1; v[1, x_]:= 1; z = 16;

u[n_, x_]:= x*u[n-1, x] + (x+1)*v[n-1, x] + 1;

v[n_, x_]:= x*u[n-1, x] + v[n-1, x] + 1;

Table[Expand[u[n, x]], {n, 1, z/2}]

Table[Expand[v[n, x]], {n, 1, z/2}]

cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];

TableForm[cu]

Flatten[%]    (* A210565 *)

Table[Expand[v[n, x]], {n, 1, z}]

cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];

TableForm[cv]

Flatten[%]    (* A210595 *)

(* Second program *)

u[n_, x_]:= u[n, x]= If[n<2, (n+1)*(1+x)^n, (1+x)*u[n-1, x] +x^2*u[n-2, x] +1+x];

T[n_]:= CoefficientList[Series[u[n, x], {x, 0, n}], x];

Table[T[n-1], {n, 12}] (* G. C. Greubel, May 23 2021 *)

PROG

(Sage)

@CachedFunction

def u(n, x): return (n+1)*(1+x)^n if (n<2) else (1+x)*u(n-1, x) + x^2*u(n-2, x) +1+x

def T(n): return taylor( u(n, x) , x, 0, n).coefficients(x, sparse=False)

flatten([T(n-1) for n in (1..12)]) # G. C. Greubel, May 23 2021

CROSSREFS

Cf. A208510, A210595.

Sequence in context: A295120 A196957 A124727 * A125101 A208519 A336725

Adjacent sequences:  A210562 A210563 A210564 * A210566 A210567 A210568

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Mar 23 2012

STATUS

approved

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Last modified May 27 16:48 EDT 2022. Contains 354110 sequences. (Running on oeis4.)